# Dispersion Decay and Scattering Theory

# Dispersion Decay and Scattering Theory

ISBN: 978-1-118-38286-8

Jul 2012

202 pages

## Description

**A simplified, yet rigorous treatment of scattering theory methods and their applications**

*Dispersion Decay and Scattering Theory* provides thorough, easy-to-understand guidance on the application of scattering theory methods to modern problems in mathematics, quantum physics, and mathematical physics. Introducing spectral methods with applications to dispersion time-decay and scattering theory, this book presents, for the first time, the Agmon-Jensen-Kato spectral theory for the Schr?dinger equation, extending the theory to the Klein-Gordon equation. The dispersion decay plays a crucial role in the modern application to asymptotic stability of solitons of nonlinear Schr?dinger and Klein-Gordon equations.

The authors clearly explain the fundamental concepts and formulas of the Schr?dinger operators, discuss the basic properties of the Schr?dinger equation, and offer in-depth coverage of Agmon-Jensen-Kato theory of the dispersion decay in the weighted Sobolev norms. The book also details the application of dispersion decay to scattering and spectral theories, the scattering cross section, and the weighted energy decay for 3D Klein-Gordon and wave equations. Complete streamlined proofs for key areas of the Agmon-Jensen-Kato approach, such as the high-energy decay of the resolvent and the limiting absorption principle are also included.

*Dispersion Decay and Scattering Theory* is a suitable book for courses on scattering theory, partial differential equations, and functional analysis at the graduate level. The book also serves as an excellent resource for researchers, professionals, and academics in the fields of mathematics, mathematical physics, and quantum physics who would like to better understand scattering theory and partial differential equations and gain problem-solving skills in diverse areas, from high-energy physics to wave propagation and hydrodynamics.

Foreword xv

Preface xvii

Acknowledgments xix

Introduction xxi

**1 Basic Concepts and Formulas 1**

1 Distributions and Fourier transform 1

2 Functional spaces 3

2.1 Sobolev spaces 3

2.2 AgmonSobolev weighted spaces 4

2.3 Operatorvalued functions 5

3 Free propagator 6

3.1 Fourier transform 6

3.2 Gaussian integrals 8

**2 Nonstationary Schrödinger Equation 11**

4 Definition of solution 11

5 Schrödinger operator 14

5.1 A priori estimate 14

5.2 Hermitian symmetry 14

6 Dynamics for free Schrödinger equation 15

7 Perturbed Schrödinger equation 17

7.1 Reduction to integral equation 17

7.2 Contraction mapping 19

7.3 Unitarity and energy conservation 20

8 Wave and scattering operators 22

8.1 Möller wave operators. Cook method 22

8.2 Scattering operator 23

8.3 Intertwining identities 24

**3 Stationary Schrödinger Equation 25**

9 Free resolvent 25

9.1 General properties 25

9.2 Integral representation 28

10 Perturbed resolvent 31

10.1 Reduction to compact perturbation 31

10.2 Fredholm Theorem 32

10.3 Perturbation arguments 33

10.4 Continuous spectrum 35

10.5 Some improvements 36

**4 Spectral Theory 37**

11 Spectral representation 37

11.1 Inversion of Fourier-Laplace transform 37

11.2 Stationary Schrödinger equation 39

11.3 Spectral representation 39

11.4 Commutation relation 40

12 Analyticity of resolvent 41

13 Gohberg-Bleher theorem 43

14 Meromorphic continuation of resolvent 47

15 Absence of positive eigenvalues 50

15.1 Decay of eigenfunctions 50

15.2 Carleman estimates 54

15.3 Proof of Kato Theorem 56

**5 High Energy Decay of Resolvent 59**

16 High energy decay of free resolvent 59

16.1 Resolvent estimates 60

16.2 Decay of free resolvent 64

16.3 Decay of derivatives 65

17 High energy decay of perturbed resolvent 67

**6 Limiting Absorption Principle 71**

18 Free resolvent 71

19 Perturbed resolvent 77

19.1 The case λ > 0 77

19.2 The case λ = 0 78

20 Decay of eigenfunctions 81

20.1 Zero trace 81

20.2 Division problem 83

20.3 Negative eigenvalues 86

20.4 Appendix A: Sobolev Trace Theorem 86

20.5 Appendix B: SokhotskyPlemelj formula 87

**7 Dispersion Decay 89**

21 Proof of dispersion decay 90

22 Low energy asymptotics 92

**8 Scattering Theory and Spectral Resolution 97**

23 Scattering theory 97

23.1 Asymptotic completeness 97

23.2 Wave and scattering operators 99

23.3 Intertwining and commutation relations 99

24 Spectral resolution 101

24.1 Spectral resolution for the Schrödinger operator 101

24.2 Diagonalization of scattering operator 101

25 T Operator and SMatrix 1003

**9 Scattering Cross Section 111**

26 Introduction 111

27 Main results 117

28 Limiting Amplitude Principle 120

29 Spherical waves 121

30 Plane wave limit 125

31 Convergence of flux 127

32 Long range asymptotics 128

33 Cross section 131

**10 Klein-Gordon Equation 133**

35 Introduction 134

36 Free Klein-Gordon equation 137

36.1 Dispersion decay 137

36.2 Spectral properties 139

37 Perturbed Klein-Gordon equation 143

37.1 Spectral properties 143

37.2 Dispersion decay 145

38 Asymptotic completeness 149

**11 Wave equation 151**

39 Introduction 152

40 Free wave equation 154

40.1 Time-decay 154

40.2 Spectral properties 155

41 Perturbed wave equation 158

41.1 Spectral properties 158

41.2 Dispersion decay 160

42 Asymptotic completeness 163

43 Appendix: Sobolev embedding theorem 165

References 167

Index 172

“The book is carefully written, features \complete and streamlined proofs", and some material, such as a novel justification of the \limiting amplitude principle", appears here for the first time.” (*Zentralblatt MATH*, 1 September 2015)